# Infinitesimal displacement in spherical coordinates

• PWiz
In summary, the conversation discusses the attempt to derive the value of ##ds^2## in spherical coordinates. The process involves breaking down the formula for Euclidean space into differential elements, simplifying it using trigonometric identities, and then rearranging the terms. There was initially a missing term, but after finding and correcting an error, the correct answer was obtained.
PWiz
I'm trying to derive what ##ds^2## equals to in spherical coordinates.
In Euclidean space, $$ds^2= dx^2+dy^2+dz^2$$
Where ##x=r \ cos\theta \ sin\phi## , ##y=r \ sin\theta \ sin\phi## , ##z=r \ cos\phi## (I'm using ##\phi## for the polar angle)
For simplicity, let ##cos \phi = A## and ##sin \phi = B##
Then ##dx=r \ cos\theta \ dB + B(cos \theta \ dr - r \ sin \theta \ d \theta)##
##dy=r \ sin\theta \ dB + B(sin \theta \ dr + r \ cos \theta \ d \theta)##
##dz= r \ dA + A \ dr##
When I add up the squares of the 3 differential elements above, gather similar terms and replace ##A## and ##B##, I get ##ds^2 = dr^2 + r^2 \ sin^2 \phi \ d \theta^2## , and you can see that one term is missing. Is something wrong with my approach?

theodoros.mihos said:
$$z = r\cos{\theta}$$
I'm using ##\phi ## as the polar angle and ##\theta## as the azimuthal angle over here. Either way, that won't add the missing term, but will only interchange the angle variables.

Sorry I see later.
$$x = rB\cos{theta} \,\Rightarrow\, dx = dr\,B\cos{\theta} + r\,dB\,\cos{theta} - rB\sin{\theta} = (B\,dr + r\,dB)\cos{\theta} - rB\sin{\theta}$$
$$dy = (B\,dr + r\,dB)\sin{\theta} +rB\cos{\theta} \,\text{and}\, dz = A\,dr-rB\,d\phi$$
because
$$A = cos{\phi} \Rightarrow dA = -\sin{\phi}\,d\phi = -B\,d\phi \,\text{and}\, dB = A\,d\phi$$
I think your problem is that A depends by B.

Last edited:
I found an error in my steps. I'm getting the correct answer now. Anyways, thanks for helping :)

## 1. What is an infinitesimal displacement in spherical coordinates?

An infinitesimal displacement in spherical coordinates refers to a very small change in position or distance in a three-dimensional space described using spherical coordinates. It is used to measure and analyze the motion or location of objects in a spherical coordinate system.

## 2. How is an infinitesimal displacement expressed in spherical coordinates?

In spherical coordinates, an infinitesimal displacement is typically expressed using three variables: radial distance (r), polar angle (θ), and azimuthal angle (φ). It can also be expressed using differential calculus notation, such as dr, dθ, and dφ, to represent small changes in these variables.

## 3. What is the significance of infinitesimal displacements in spherical coordinates?

Infinitesimal displacements in spherical coordinates are important in understanding the motion and position of objects in a three-dimensional space, particularly in physics and engineering. They allow for precise calculations and analysis of motion and changes in position.

## 4. How are infinitesimal displacements used in calculus?

In calculus, infinitesimal displacements are used to calculate derivatives and integrals in spherical coordinates. They are also used in differential equations to model and solve problems involving motion and change in position in a three-dimensional space.

## 5. Can infinitesimal displacements be used in other coordinate systems?

Yes, infinitesimal displacements can be used in other coordinate systems, such as Cartesian coordinates or cylindrical coordinates. However, the expressions and calculations for infinitesimal displacements may differ depending on the coordinate system being used.

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